Positive Element
In mathematics, especially functional analysis, a self-adjoint (or hermitian) element of a C*-algebra is called positive if its spectrum consists of non-negative real numbers. Moreover, an element of a C*-algebra is positive if and only if there is some in such that . A positive element is self-adjoint and thus normal.
If is a bounded linear operator on a Hilbert space, then this notion coincides with the condition that is non-negative for every vector in . Note that is real for every in if and only if ' is self-adjoint. Hence, a positive operator on a Hilbert space is always self-adjoint (and a self-adjoint everywhere defined operator on a Hilbert space is always bounded because of the Hellinger-Toeplitz theorem).
The set of positive elements of a C*-algebra forms a convex cone.
Read more about Positive Element: Positive and Positive Definite Operators, Examples, Partial Ordering Using Positivity
Famous quotes containing the words positive and/or element:
“If all political power be derived only from Adam, and be to descend only to his successive heirs, by the ordinance of God and divine institution, this is a right antecedent and paramount to all government; and therefore the positive laws of men cannot determine that, which is itself the foundation of all law and government, and is to receive its rule only from the law of God and nature.”
—John Locke (1632–1704)
“We must be very suspicious of the deceptions of the element of time. It takes a good deal of time to eat or to sleep, or to earn a hundred dollars, and a very little time to entertain a hope and an insight which becomes the light of our life.”
—Ralph Waldo Emerson (1803–1882)